| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) |
| Year | 2012 |
| Session | June |
| Marks | 12 |
| Topic | Impulse and momentum (advanced) |
| Type | Find coefficient of restitution |
| Difficulty | Challenging +1.8 This is an advanced 2D collision problem requiring simultaneous application of conservation of momentum (in two perpendicular directions), Newton's experimental law, and energy considerations. It demands setting up a system of equations with multiple unknowns, algebraic manipulation involving the constraint that final velocities are perpendicular, and solving for both the coefficient of restitution and an angle. This goes well beyond standard 1D collision exercises and requires sophisticated problem-solving across multiple mechanics principles. |
| Spec | 6.03b Conservation of momentum: 1D two particles6.03i Coefficient of restitution: e6.03j Perfectly elastic/inelastic: collisions |
**Question 10(i)**
Let $u$ denote speed of sphere $Q$ before impact, $v_1$ and $v_2$ the speeds of spheres $Q$ and $P$, respectively, after impact and $\alpha$ the angle between $Q$'s initial direction of motion and the line of centres.
After impact, if moving perpendicularly, $Q$ moves perpendicular to line of centres and $P$ moves along line of centres. (Stated or implied.) **B1**
CLM: $mu\cos\alpha = 0 + 3mv_2$ or $mu_x = 3mv$ **M1A1**
NEL: $eu\cos\alpha = v_2$ or $eu_x = v$. **A1**
$\therefore e = \frac{1}{3}$ **A1** [5]
**Question 10(ii)**
$v_1 = u\sin\alpha$ and $v_2 = \frac{1}{3}u\cos\alpha$. **B1 (both)**
Loss in KE is
$\frac{1}{2}mu^2 - \frac{1}{2}mu^2\sin^2\alpha - \frac{1}{2} \cdot 3m\frac{u^2\cos^2\alpha}{9}$ **M1A1**
$= \frac{1}{12}mu^2$ (Or remaining KE is 5/6 of initial KE etc.) **A1**
But $\cos^2\alpha + \sin^2\alpha = 1$ (used) **M1**
$\Rightarrow \ldots \Rightarrow \sin^2\alpha = \frac{3}{4} \Rightarrow \sin\alpha = \frac{\sqrt{3}}{2} \Rightarrow \alpha = 60°$ **M1A1** [7] **[12]**10\\
\includegraphics[max width=\textwidth, alt={}, center]{d8ca5464-435f-45e0-8e19-1830415a7c60-5_432_949_258_598}
A smooth sphere $P$ of mass $3 m$ is at rest on a smooth horizontal table. A second smooth sphere $Q$ of mass $m$ and the same radius as $P$ is moving along the table towards $P$ and strikes it obliquely (see diagram). After the collision, the directions of motion of the two spheres are perpendicular.\\
(i) Find the coefficient of restitution.\\
(ii) Given that one sixth of the original kinetic energy is lost as a result of the collision, find the angle between the initial direction of motion of $Q$ and the line of centres.
\hfill \mbox{\textit{Pre-U Pre-U 9795/2 2012 Q10 [12]}}